Accelerated Math III
Unit 5: Investigating Trigonometry Graphs
5.1: What’s Your Temperature? Learning Task:
Scientists are continually monitoring the average temperatures across the globe to determine if Earth is
experiencing Climate Change. One statistic scientists use to describe the climate of an area is average
temperature. The average temperature of a region is the mean of its average high and low temperatures.
1.
The graph to the right shows the average high and low temperature in Atlanta from January to December. The
average high temperatures are in red and the average
low temperatures are in blue.
a.
How would you describe the climate of Atlanta,
Georgia?
b. If you wanted to visit Atlanta, and prefer average
highs in the 70’s, when would you go?
c.
Estimate the lowest and highest average high temperature. When did these values occur?
d. What is the range of these temperatures?
e.
Estimate the lowest and highest average low temperature. When did these values occur?
f.
What is the range of these temperatures?
2. In mathematics, a function that repeats itself in regular intervals, or periods, is called periodic.
a. If you were to continue the temperature graphs above, what would you consider its interval, or period, to
be?
b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods.
.
c.
What function have you graphed that looks similar to this graph?
3. How do you think New York City’s averages would compare to Atlanta’s?
4. Use the data in the table below to create a graph for Sydney, Australia’s average high and low temperatures.
Avg high
Jan
Feb
March
April
May
June
July
August
Sept
Oct
Nov
Dec
78
79
77
73
67
64
62
65
69
72
74
77
67
68
65
60
55
51
49
50
54
58
61
65
temp F
Avg low
temp F
Use the graph to compare Sydney’s climate to Atlanta’s. What do you notice?
5. Sine and Cosine functions can be used to model average temperatures for cities. Based on what you know about
these graphs from earlier units, why do you think these functions are more appropriate than a cubic function?
Or an exponential function?
6. Using the data from #4, use your graphing calculator to find a sine function that models the data. Record your
functions here.
a.
Average high temp:
b. Average low temp:
The a, b, c and d values your calculator reported have specific effects upon the sin graph. In the next task you will
investigate the characteristics of sine, cosine and tangent graphs. You will then revisit the functions in #6 to
determine the effects of the values a – d.
5.2: Exploring Sine and Cosine Graphs
In the previous task you used your calculator to model periodic data using a sine graph. Now you will explore the
sine, cosine, and tangent graphs to determine the specific characteristics of these graphs.
1.
Using your knowledge of the unit circle, complete the following chart for f(x)=sin x. (Use exact values.)
x
-2π
−
7𝜋
4
−
3𝜋
2
−
5𝜋
4
-π
−
3𝜋
4
−
𝜋
2
−
𝜋
4
0
𝜋
4
𝜋
2
3𝜋
4
π
5𝜋
4
3𝜋
2
7𝜋
4
2π
sin x
a. What do you notice about the values in the chart?
b. When is sin x=1? 0?
c. Does sin x appear to be a periodic function? If so, at what would you consider to be its period?
2. Use your graphing calculator to graph sin x. Check your mode to make sure you are using radians and make sure
you have an appropriate window for your data. Make a grid below and draw an accurate graph of sin x. Make
sure you draw a smooth curve.
3. Study the graph to answer the following questions:
a.
What is the period?
b. What is the domain and range?
c.
What is the y-intercept?
d. Where do the x-intercepts occur?
e.
What are the maximum values and where do they occur?
f.
What is the minimum value and where does it occur?
g.
How would your answers to questions d, e, and f change if your graph continued past 360?
4. Using your knowledge of the unit circle and radians, complete the following chart for f(x)=cos x
x
-2π
−
7𝜋
4
−
3𝜋
2
−
5𝜋
4
-π
−
3𝜋
4
−
𝜋
2
−
𝜋
4
0
𝜋
4
𝜋
2
3𝜋
4
π
5𝜋
4
3𝜋
2
7𝜋
4
2π
cos x
5. Use your graphing calculator to graph cos x. Check your mode to make sure you are using radians and make sure
you have an appropriate window for your data. Make a grid below and draw an accurate graph of cos x. Make
sure you draw a smooth curve.
6. Study the graph to answer the following questions:
a.
What is the period?
b. What is the domain and range?
c.
What is the y-intercept?
d. Where do the x-intercepts occur?
e. What is the maximum value and where does it occur?
f.
What is the minimum value and where does it occur?
g.
Picture your graph continuing past 2π. How would your answers to questions d, e, and f change?
h. Re-write your answer to d, e and f and make sure to include all possible values.
i.
Using your graph, find the value of cos
9π
j.
Find all values of θ, such that cos 𝜃 = −
√2
.
7.
4
.
2
Using your calculator, graph sin x and cos x on the same axis. How are they alike? How are they different?
8. Using what you have learned about the graphs of the sine and cosine functions, practice graphing the functions
by hand. You should be able to quickly sketch the graphs of these functions, making sure to include zero’s,
intercepts, maximums and minimums. Be sure to create a smooth curve!
The graphs of trigonometric functions can be transformed in ways similar to the function transformations you
have studied earlier.
9. Using your knowledge of transformation of functions make a conjecture about the effect 2 will have on the
following graphs of cos x and sin x. Then graph it on a calculator to determine if your conjecture is correct.
Graph the parent function each time to compare the effect of 2 on the graph. (Make sure your mode is set on
radians.)
a.
f(x)= 2sin x
f(x)= 2cos x
b.
f(x)= sin x + 2
f(x)= cos x +2
c.
d.
f(x)= sin (x - 2)
(x)= sin (2x)
f(x)= cos (x -2)
f(x)= cos (2x)
Using your conjectures from above, sketch the graph of these functions. Use your calculator to check your
graphs.
e.
f(x)=2 sin(x)+2
f(x)= 2cos(2x)
Consider the functions f(θ)=A sin(kθ + c)+h and f(θ)=A cos(kθ + c)+h. A, k, c and h have specific effects on the
graphs of the function. In trigonometry we also have special names for them based on their effect on the graph.
10. The amplitude of the function is |A|. Look back at 9a: f(x)= 2sin x and f(x)= 2cos x
a.
What is the value of A? How is it related to your graphs?
b. How is the amplitude related to the distance between the maximum and minimum values?
c.
What effect does A have on the effect if A<0? Graph y=-2sin x to test your conjecture.
11. The period of the function is related to the value of k. Look at 9d: f(x)=sin(2x) f(x)=cos(2x)
The period of the sine and cosine function is defined as
a.
𝟐𝝅
𝒌
, where k>0.
When x was multiplied by 2, the function repeated itself twice in the usual period of 2π. What is the period
of the functions in 9d?
b. Using your calculator look at the graph of the functions f(x)=sin x and f(x)=sin(4x).
Notice the graph repeats 4 times in the length of time it took sin x to complete one period. So the period of
f(x)=sin(4x) is
𝟐𝝅
𝟒
𝝅
or .
𝟐
12. Look back at 9c. The 2 shifted the graph 2 units to the left. A horizontal translation of a trigonometric
function is called a phase shift. When calculating the phase shift you have to consider the value of k, which
𝒄
determines the period of the function. The phase shift = − .
𝒌
If c>0, the graph shifts to the left.
If c>0, the shift is to the right.
a.
Graph f(x) = sin x and g(x)=sin(x+2)on the same axis using radians.
For what values of x is f(x) = 0?
For what values of x is g(x) = 0?
b. What do you notice about your answers to II and III?
c.
Use the table feature on your calculator to investigate this relationship for values other than f(x)=0.
13. Graph y=sin x and y= cos x on the same axes. How can you use the sine function to match the graph of the
cosine function? How can you use the sine function to match the graph of the cosine function?
14. In trigonometric functions, h translates the graph h units vertically.
a. What effect does 2 have on the graph of the function y = f(x) + 2?
b. What effect did the 2 have on the graphs for 9b?
c.
What happens when h > 0? h < 0 ? h = 0?
15. Look at this equation that models the average monthly temperatures for Asheville, NC. (The average monthly
temperature is an average of the daily highs and daily lows.)
𝜋
Model for Asheville, NC 𝑓(𝑡) = 18.5 sin ( 𝑡 − 4) + 54.5 where t = 1 represents January
6
a.
Find the values of A, k, c, and h in the equation.
b. Graph the equation on your calculator. The maximum and minimum values of a periodic function oscillate
about a horizontal line called the midline. What is the midline of the equation modeling Asheville’s
temperature?
c.
How it the value of the amplitude related to this midline?
16. How is the amplitude related to the midline of f(x)=sin x? f(x)=3sinx + 2?
f(x)=-4sin x -2?
17. State the amplitude and period of the following functions describe the graph of the function.
a.
b.
f(θ)=2sin(6θ)
A = __ period = ___
f(θ)= -4cos(1/2 θ)
A = _______ period = __
The frequency of a sine or cosine function refers to the number of times it repeats compared to the parent
function’s period. Frequency is usually associated with the unit Hertz or oscillations/second and measures the
number of repeated cycles per second. The frequency and period are reciprocals of each other.
Example: Determine the equation of a cosine function that has a frequency of 4.
1
The frequency and period are reciprocals of each other so the period = .
4
Since
If
, we can replace the period with
, then
.
The cosine function with a frequency of 4 is f(θ)=cos(8πθ) .
1
4
and solve for k.
18. Find the equation of a sine graph with a frequency of 6 and amplitude of 4.
5.3: Applications of Sine and Cosine Graphs
The Ferris Wheel
There are many rides at the amusement park whose movement can be described using trigonometric functions. The
Ferris Wheel is a good example of periodic movement.
Sydney wants to ride a Ferris wheel that has a radius of 60 feet and is
suspended 10 feet above the ground. The wheel rotates at a rate of 2
revolutions every 6 minutes. (Don’t worry about the distance the seat
is hanging from the bar.) Let the center of the wheel represents the
origin of the axes.
a. Write a function that describes a Sydney’s height above the
ground as a function of the number of seconds since she was ¼ of
the way around the circle (at the 3 o’clock position).
b. How high is Sydney after 1.25 minutes?
c. Sydney’s friend got on after Sydney had been on the Ferris wheel long enough to move a quarter of the way around
the circle. How would a graph of her friend’s ride compare to the graph of Sydney’s ride? What would the equation
for Sydney’s friend be?
*There are additional activities in this task that are available on request.
5.4: Graphing Other Trigonometric Functions
You have spent time working with Sine and Cosine graphs and equations. You will now have a chance to move beyond
those functions into related functions.
1.
Think back to right triangle trigonometry. How was the tangent function defined?
2. Solve the following problem. (Hint: Make sure to draw a picture.)
A light-house keeper standing 50 feet above the ground sees a boat in distress at a 15 angle of depression.
How far is the boat from the base of the lighthouse?
3. There is also a mathematical definition of tangent as related to circles. List everything you know about the
tangent of a circle.
4. In the figures below segment TP is perpendicular to segment ON. Line CN is tangent to circle O at T. N is the
point where the line intersects the x-axis and C is the where the line intersects the y-axis. (Only segment CN
is shown.)
Use your knowledge of sine and cosine to determine the length of segment TN. Use exact answers, no decimal
approximations.
5. Using your understanding of the unit circle and 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 =
angles.
θ
Sin θ
Cos θ
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
, to complete the chart below for the indicated
Tan θ
Cotangent θ
30
45
60
How are these values of tangent related to the length you found in #4?
What would happen to the length of TN if the angle was changed to 0?
What would happen to the length of TN if the angle was changed to 90?
6. Complete the chart below using the unit circle. Remember, 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 =
x
−
3𝜋
2
−
5𝜋
4
-π
−
3𝜋
4
−
𝜋
2
−
𝜋
4
0
𝜋
4
𝜋
2
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
.
3𝜋
4
π
5𝜋
4
tan x
How do you indicate an undefined solution for specific points in the domain of a graph?
3𝜋
2
Use your graph and chart to answer the following questions.
a.
What is the period?
b. What is the domain and range?
c.
What is the y-intercept?
d. Where do the x-intercepts occur?
e.
Does the graph have any maximum or minimum values? If so, what are they?
f.
Does the graph have any asymptotes? If so, where are they?
7. Use your graph to find the following values of tangent.
7𝜋
a. tan(-4π)=
b. tan =
c.
2
Tan
−8𝜋
4
d.
=
tan
−9𝜋
4
=
8. Go back to the drawings in problem #4. Solve for the values of segment CT.
9. The cotangent is the reciprocal of the tangent. So cotangent θ =
1
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃
the chart in #5 by adding cotangent in the blank column beside tangent.
. Using this definition, complete
How are the values of cotangent in the chart related to the length of segment CT from problem #8?
What do you notice about the values of the tangent and cotangent values in the chart?
10. Complete the chart below and then graph the cotangent on the given grid.
x
-π
−
3𝜋
4
−
𝜋
2
−
𝜋
4
0
𝜋
4
𝜋
2
3𝜋
4
π
Cotangent
x
How do you indicate an undefined solution for specific points in the domain of a graph?
Use your graph and chart to answer the following questions.
a.
What is the period?
b. What is the domain and range?
c.
What is the y-intercept?
d. Where do the x-intercepts occur?
g.
Does the graph have any maximum or minimum values? If so, what are they?
h. Does the graph have any asymptotes? If so, where are they?
11. Use your graph to find the following values of cotangent.
a.
cot(-5π)=
b. Cot
9𝜋
2
c.
=
Cot
−5𝜋
4
=
d.
cot(0)=
The cosecant and secant functions are reciprocals of the sine and cosine functions. The cosecant, csc, is the
reciprocal of the sine function. The secant, sec, function is the reciprocal of the cosine function.
csc 𝜃 =
1
sin 𝜃
sec θ =
1
cos 𝜃
These graphs will be explored using a graphing calculator.
12. Graph the sine and cosecant function.
Enter the sine function in y1.
Check your window and mode to make sure there are at least two periods of the graph on your screen- one
in the 1st quadrant and one in the 2nd quadrant.
Enter the cosecant function in y2.
The cosecant function may need to be entered as
a.
1
sin 𝜃
.
What are the values of sin x at x = -2π, - π, 0, π, and 2π?
b. What are the values of csc x at x = -2π, - π, 0, π, and 2π? Why?
c.
What are the values of x where the graphs are tangent?
Answer the following questions for csc x.
d. What is the period?
e.
What is the domain and range?
f.
What is the y-intercept?
g.
Where do the x-intercepts occur?
i.
Does the graph have any maximum or minimum values? If so, what are they?
j.
Does the graph have any asymptotes? If so, where are they?
13. Graph the cosine and secant function.
Enter the cosine function in y1.
Check your window and mode to make sure there are at least two periods of the graph on your screen- one
in the 1st quadrant and one in the 2nd quadrant.
Enter the secant function in y2.
The secant function may need to be entered as
a.
1
sin 𝜃
.
What are the values of cos x at x = -3π/2, - π/2, π/2, and 3π/2?
b. What are the values of sec x at x = x = -3π/2, - π/2, π/2, and 3π/2? Why?
c.
What are the values of x where the graphs are tangent?
Answer the following questions for sec x.
e.
What is the period?
h. What is the domain and range?
i.
What is the y-intercept?
j.
k.
Where do the x-intercepts occur?
Does the graph have any maximum or minimum values? If so, what are they?
l.
Does the graph have any asymptotes? If so, where are they?
14. Which of the following functions have periods of 2π? Which have periods of π?
sin x, cos x, tan x, cot x, csc x, sec x
.
The transformations of the tangent, cotangent, secant, and cosecant functions work the same way as the
transformations of the sine and cosine. The only difference is the period of the tangent and cotangent.
The period of functions y = sin kθ, y = cos kθ, y = csc kθ, and y = sec kθ is 2π/k.
The period of functions y = tan kθ, and y = cot kθ is π/k.
15. Write an equation for the indicated function given the period, phase shift and vertical translation.
a. Tangent function: period = 2π, phase shift = π, and vertical shift = -3
b. Secant function: period = π, phase shift = -π, and vertical shift = 4
c.
Cosecant function: period = 4π, phase shift = -π/2, and vertical shift = -6
16. Graph the following functions.
a.
f(θ) = tan (θ – π/4) + 2
b. f(θ) = cot (θ – 3π/2) – 1
c.
f(θ) = sec (2θ + π/2) – 1 (Hint: sketch f(θ ) = cos (2θ + π/2) – 1)
𝜃
d. 𝑓(𝜃) = csc( − 𝜋) − 3
2
𝜃
(Hint: sketch 𝑓(𝜃) = sin( − 𝜋) − 3)
2
17. Let’s take one more look at your figures from problem 4. Find the lengths OC and ON.
What connection can you make between these values and the cosecant and secant values for the given angles?
5.5: Inverses of Trigonometric Functions
In an earlier task you looked at this function that models the average monthly temperatures for Asheville, NC.
(The average monthly temperature is an average of
Model for Asheville, NC

6
the daily highs and daily lows.)
f(t)=18.5sin( t - 4)+ 54.5 where t=1 represents January.
How can you use this model to find the month that has a specific average temperature?
Recall your work with inverse relationships. The inverse of a function can be found by interchanging the
coordinates of the ordered pairs of the function. In this case, the ordered pair (month, temperature) would
become (temperature, month).
This task will allow you to explore the inverses the trigonometric functions from a geometric and algebraic
perspective.
1.
Graph 𝑓(𝜃) = sin 𝜃 and the line y = ½ .
a.
How many times do these functions intersect between -2π and 2π?
b. How is this graph related to finding the solution to ½ = sin θ?
c.
If the domain is not limited, how many solutions exist to the equation ½ = sin θ?
d. Would this be true for the other trigonometric functions? Explain.
2. What happens to the axes and coordinates of a function when you reflect it over the line y = x?
a.
Sketch a graph of 𝑓(𝜃) = cos 𝜃 and its reflection over the line y = x.
b. How many times does the line x= ½ intersect the reflection of cos x?
3.
Sketch the inverse of 𝑓(𝜃) = tan 𝜃 and determine if it is a function.
𝜋
𝜋
2
2
4. Look at this graph of 𝑓(𝜃) = sin 𝜃 with domain − ≤ 𝜃 ≤
and its inverse.
Is the inverse a function now?
5. Use the following graphs to determine the limited domains on the cosine function used to insure the inverse is a
function.
Highlight the axes that represent the angle measure. (on
both
graphs)
6. Use
the
following
graphs
to
determine the limited domains on the tangent function used to insure the inverse is a function.
that represent the angle measure.
Mark the axes
7. We use the names sin-1, cos -1 , and tan-1 or arcsin, arccos, and arctan to represent the inverse of these
functions on the limited domains you explored above. The values in the limited domains of sine, cosine and
tangent are called principal values. (Similar to the principal values of the square root function.) Calculators
give principal values when reporting sin-1, cos -1 , and tan-1.
Complete the chart below indicating the domain and range of the given functions.
Function
Domain
Range
𝑓(𝜃) = sin−1 𝜃
𝑓(𝜃) = cos −1 𝜃
𝑓(𝜃) = tan−1 𝜃
The inverse functions do not have ranges that include all 4 quadrants. Add a column to your chart that indicates
the quadrants included in the range of the function. This will be important to remember when you are determining
values of the inverse functions.
8. Use what you know about trigonometric functions and their inverses to evaluate the following expressions. Two
examples are included for you. (Unit circles can also be useful.)
Example 1:
Example 2:
√3
ArcCos ( )
The answers will be
2
sin (cos-1 1 + tan-1 1)
The answer will be a
number, not an angle.
an angle.
Simplify parentheses
first.
√3
Let θ = Arccos ( ) Ask yourself, what
2
angle has a cos
value of
√3
Cos θ = ( )
√3
2
.
Using the definition
2
of Arccos.
𝜋
5𝜋
θ=( )
Why isn’t ( )
6
θ = cos-1 1
θ = 0o
θ = tan-1 1
θ = 45
sin (0o + 45o)
sin(45o) =
Substitution
√2
2
6
included?
So,
So,
√3
𝜋
2
6
ArcCos ( ) = ( )
sin (cos-1 1 + tan-1 1) =
√2
2